Biomedical Engineering Reference
In-Depth Information
Fig. 2.2
Swelling pressure vs. thickness by Hedbys and Dohlman (
1963
) and Olsen and Sper-
ling (
1987
), the Donnan model and the Poisson-Boltzmann (PB) model. Data plotted on the
normal axis (
a
) and a log-log axis (
b
). Fitting curve by Olsen:
P
s
t
−
3
.
48
=
7
.
56
×
mmHg and
the fitting curve based on Hedbys' data (Fatt,
1968
)
P
s
H)
mmHg. The hydra-
tion data from Hedbys and Dohlman (
1963
) is transformed to thickness by the linear relation
H
=
7
.
00
t
−
0
.
64 (Hedbys and Mishima,
1966
). Charge density for Donnan model and the PB
model is
ρ
0
/F
=
48 mM
=
1810
×
exp
(
−
A
s
immersed in an ionic solution and constrained by a porous piston to a thickness
t
.
The total Gibbs free energy of the system is defined as
G
=
F
+
PV
+
P
s
A
s
t,
(2.6)
where
is the Helmholtz free energy,
P
and
V
are the atmosphere pressure and
total system volume, respectively, and
P
s
is the pressure exerted by the piston on the
sample (Katchalsky and Michaeli,
1955
; Hart and Farrell,
1971
). At equilibrium we
require
F
1
A
s
∂
∂t
(
P
s
=−
F
+
PV).
(2.7)
At fixed temperature
T
and atmosphere pressure
P
,the
PV
term may be dropped
and the swelling pressure is then expressed as
T,P
=−
T,P
1
A
s
∂
∂t
∂
∂V
t
P
s
=−
,
(2.8)
which is simply the derivative of the Helmholtz free energy with respect to the tissue
volume
V
t
.
In general, the Helmholtz free energy
F
will have electrostatic
F
el
and chemo-
mechanical components
F
cm
(Jin and Grodzinsky,
2001
). The effective electrostatic
free energy of a polyelectrolyte solution in a mean-field approximation can be ex-
pressed as a functional of the electrostatic potential
ϕ
and local ionic concentrations
